In each question two equations numbered (I) and (II) are given. You should solve both the equations and mark appropriate answer
I. ` 4 x^(2) - 8 x - 5 = 0 `
II. ` 2 y ^(2) - 11 y + 14 = 0`

Options

A. If x= y or no relation
B. If ` x gt y `
C. If ` x lt y `
D. If `x ge y`

Answer

To solve the given equations step by step, we will start with the first equation (I) and then proceed to the second equation (II).

### Step 1: Solve Equation I
The first equation is:
\[ 4x^2 - 8x - 5 = 0 \]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a = 4 \), \( b = -8 \), and \( c = -5 \).

#### Step 1.1: Calculate the Discriminant
The discriminant \( D \) can be calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-8)^2 - 4 \cdot 4 \cdot (-5) \]
\[ D = 64 + 80 \]
\[ D = 144 \]

#### Step 1.2: Find the Roots Using the Quadratic Formula
The roots can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-8) \pm \sqrt{144}}{2 \cdot 4} \]
\[ x = \frac{8 \pm 12}{8} \]

Calculating the two possible values for \( x \):
1. \( x_1 = \frac{8 + 12}{8} = \frac{20}{8} = 2.5 \)
2. \( x_2 = \frac{8 - 12}{8} = \frac{-4}{8} = -0.5 \)

### Step 2: Solve Equation II
The second equation is:
\[ 2y^2 - 11y + 14 = 0 \]

This is also a quadratic equation where \( a = 2 \), \( b = -11 \), and \( c = 14 \).

#### Step 2.1: Calculate the Discriminant
Using the discriminant formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-11)^2 - 4 \cdot 2 \cdot 14 \]
\[ D = 121 - 112 \]
\[ D = 9 \]

#### Step 2.2: Find the Roots Using the Quadratic Formula
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ y = \frac{-(-11) \pm \sqrt{9}}{2 \cdot 2} \]
\[ y = \frac{11 \pm 3}{4} \]

Calculating the two possible values for \( y \):
1. \( y_1 = \frac{11 + 3}{4} = \frac{14}{4} = 3.5 \)
2. \( y_2 = \frac{11 - 3}{4} = \frac{8}{4} = 2 \)

### Step 3: Compare the Values of x and y
We have the following values:
- From Equation I: \( x_1 = 2.5 \), \( x_2 = -0.5 \)
- From Equation II: \( y_1 = 3.5 \), \( y_2 = 2 \)

Now we compare:
1. \( 2.5 \) (from \( x_1 \)) and \( 3.5 \) (from \( y_1 \)): \( 2.5 < 3.5 \)
2. \( 2.5 \) (from \( x_1 \)) and \( 2 \) (from \( y_2 \)): \( 2.5 > 2 \)
3. \( -0.5 \) (from \( x_2 \)) and \( 3.5 \) (from \( y_1 \)): \( -0.5 < 3.5 \)
4. \( -0.5 \) (from \( x_2 \)) and \( 2 \) (from \( y_2 \)): \( -0.5 < 2 \)

### Conclusion
From the comparisons, we can see:
- \( x_1 = 2.5 \) is less than \( y_1 = 3.5 \)
- \( x_1 = 2.5 \) is greater than \( y_2 = 2 \)
- \( x_2 = -0.5 \) is less than both \( y_1 = 3.5 \) and \( y_2 = 2 \)

Thus, we cannot establish a consistent relationship between \( x \) and \( y \) across all comparisons. Therefore, we conclude that there is no definitive relationship between \( x \) and \( y \).

### Final Answer
No relation can be determined between \( x \) and \( y \).

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